On the choice of signs for householder's matrices

摘要

The numerical construction of Householder's matrices of the form I-2wwH is known to be a problem with two distinct solutions; more precisely, the actual construction of such a matrix in a given context involves a choice of sign, and it is widely believed that only one alternative is correct, the other one leading to possible numerical unstabilities. This paper shows that the numerical stability of the process depends not on the chosen sign itself but only on the implementation of the actual computations; as well-conditioned approach for the non-classical case is presented and illustrated by a numerical example. Both signs are thus equally correct and there seems to be no reason at all why a specific sign should be prefered to the other.